On a covering property of rarefied sets at infinity in a cone. (English) Zbl 1119.31003

Aikawa, Hiroaki (ed.) et al., Potential theory in Matsue. Selected papers of the international workshop on potential theory, Matsue, Japan, August 23–28, 2004. Tokyo: Mathematical Society of Japan (ISBN 4-931469-33-7/hbk). Advanced Studies in Pure Mathematics 44, 233-244 (2006).
This paper is concerned with domains of the form \(C(\Omega )=\{rz:r>0,z\in \Omega \}\), where \(\Omega \) is a \(C^{2,\alpha }\)-domain in the unit sphere of \(\mathbb{R}^{n}\). There is a Martin function for \(C(\Omega ) \), associated with the point at infinity, which has the form \(K(rz)=r^{\alpha }f(z)\) for \( r>0\) and \(z\in \Omega \). If \(v\) is a positive superharmonic function on \( C(\Omega ) \) such that \(\inf_{C(\Omega )}v/K=0\), then the set \(E_{v}\), defined as \(\{rz\in C(\Omega ):v(rz)\geq r^{\alpha }\}\), cannot be large near infinity. The authors show that such a set \(E_{v}\) can always be covered by a sequence of balls \(B(x_{k},r_{k})\) such that \(\sum (r_{k}/\left| x_{k}\right| )^{n-1}<\infty \). The case of a halfspace had previously been treated by V. S. Azarin [Mat. Sb. (N.S.) 66 (108), 248–264 (1965; Zbl 0135.32203)] and M. Essén, H. L. Jackson and P. J. Rippon [Hiroshima Math. J. 15, 393–410 (1985; Zbl 0594.31014)].
For the entire collection see [Zbl 1102.31001].


31B25 Boundary behavior of harmonic functions in higher dimensions
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions